Can all surfaces be turned inside out? I saw this YouTube video claiming that spheres, double torues, triple toruses, etc. can all be turned inside out, but what about other surfaces? Are there any surfaces that can't be turned inside out?
 A: In mathematical terms, your question may be restated as "does every closed orientable manifold of dimension two admit an orientation reversing diffeomorphism?" As noted by others, the question does not make sense for non orientable surfaces, and so we have to limit ourselves to orientable spaces. 
A key phrase here is the classification of surfaces, which puts surfaces into three categories: the sphere, tori with several holes (surfaces of genus $g$), and several copies of the projective plane glued together. It can be shown that any closed $2$-manifold must be of this type. So in short yes, since the orientable surfaces are precisely the tori with several holes, all orientable surfaces admit an orientation reversing diffeomorphism. 
It should be noted that the answer is negative for higher dimensional orientable spaces. For example, even dimensional complex projective spaces do not admit orientation reversing diffeomorphisms. 
Edit: I am limiting myself here to closed orientable surfaces. Non-closed surfaces are not as nice and I'm not sure there's a clear answer, perhaps someone else can fill in those details. 
A: For the class of surfaces called non-orientable, there's no such thing as inside or out.
This class includes the famous Moebius Strip and Klein Bottle, for instance.
A: A Klein bottle is a surface with no inside and no outside. So in that case I don't think it would make any sense to talk about turning it inside out.
